422 RICE 



ART. K 



length and the latter the square.) It follows that if equilibrium 

 exists the component of force across the surface QRS parallel 

 to OX is, for a small value of K, practically equal to 



K{aXx + pXr + yXz). 



The quotient of this force by the area K is the a:-component of 

 the stress at P across the plane (a, /3, 7) (meaning the plane 

 whose normal has these direction cosines). Similar results can 

 be obtained for the other components, and we arrive at the result 

 that the stress across the plane {a, /S, 7) has the components 



aXx + pXy + yXz, aYx + ^Yr + yVz, 



aZx + /3Zk + yZz. 



(28) 



We know that in fluid media in equilibrium the pressure 

 varies with the depth owing to the action of gravity, and in 

 general the pressure at a point varies with the position of the 

 point when body forces are exerted on the fluid. The reader 

 may be acquainted with the relation between the "gradient of the 

 pressure" (i.e., the rate of variation of pressure per unit of dis- 

 tance in a given direction) and the body force. It is dealt with 

 in works on hydromechanics and is given by the equations 



dx dy dz 



where Fx, Fy, Fz are the components of the force F on unit 

 volume of the fluid. Moreover, if at any point on the surface 

 of the fluid there is an external force in the nature of a thrust or 

 pull on the surface, and if F is the value of it per unit surface at 

 the point, then the value of the pressure at that point of the 

 surface is given by 



ap = -F,, /3p = -Fy, 7p = -F^, 



where a, /3, 7 are the direction cosines of the outwardly directed 

 normal to the surface at the point. By exactly the same type 

 of reasoning which leads to this result, we can find relations 

 between the body forces on a solid body and the space differ- 

 ential coefficients of the "stress constituents" Xx, Xy, . . • Zz. 



